{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:CTNXVNR55V2SAHJO33PKRUFG64","short_pith_number":"pith:CTNXVNR5","canonical_record":{"source":{"id":"1508.04918","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-08-20T08:28:55Z","cross_cats_sorted":[],"title_canon_sha256":"b59db81c6aac22f446ceadede859f0563e756d80ff94e79732086958f81615f8","abstract_canon_sha256":"4fb24aa8e40c893833889b078ce3e8207292f068e911c69bef6aa2e7c213c24e"},"schema_version":"1.0"},"canonical_sha256":"14db7ab63ded75201d2ededea8d0a6f739ea9f92b14a18851196708c8c5d6b0b","source":{"kind":"arxiv","id":"1508.04918","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1508.04918","created_at":"2026-05-18T01:18:43Z"},{"alias_kind":"arxiv_version","alias_value":"1508.04918v2","created_at":"2026-05-18T01:18:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1508.04918","created_at":"2026-05-18T01:18:43Z"},{"alias_kind":"pith_short_12","alias_value":"CTNXVNR55V2S","created_at":"2026-05-18T12:29:17Z"},{"alias_kind":"pith_short_16","alias_value":"CTNXVNR55V2SAHJO","created_at":"2026-05-18T12:29:17Z"},{"alias_kind":"pith_short_8","alias_value":"CTNXVNR5","created_at":"2026-05-18T12:29:17Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:CTNXVNR55V2SAHJO33PKRUFG64","target":"record","payload":{"canonical_record":{"source":{"id":"1508.04918","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-08-20T08:28:55Z","cross_cats_sorted":[],"title_canon_sha256":"b59db81c6aac22f446ceadede859f0563e756d80ff94e79732086958f81615f8","abstract_canon_sha256":"4fb24aa8e40c893833889b078ce3e8207292f068e911c69bef6aa2e7c213c24e"},"schema_version":"1.0"},"canonical_sha256":"14db7ab63ded75201d2ededea8d0a6f739ea9f92b14a18851196708c8c5d6b0b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:18:43.626590Z","signature_b64":"csjovNKj/maXTYACiPdxEROumBtf6bNYvqYaQjP98QoFaFKSTc9n7Kw9gLtLuoZ86bRm4Um/04tTkdexS1sPCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"14db7ab63ded75201d2ededea8d0a6f739ea9f92b14a18851196708c8c5d6b0b","last_reissued_at":"2026-05-18T01:18:43.625535Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:18:43.625535Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1508.04918","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:18:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"K8F0Alwhaxf5kJGfrAbnHO3fqOe5Mej0iCnK0nOmSyYKpCRJoAJbaB4sGZiNgUhehwztTQ1ZHIIVOGbQrQwqBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T14:42:12.118509Z"},"content_sha256":"fb7d80fd937c1bb506cfc49dda883b69b8963f800e57057130d427bb0b5eb3cf","schema_version":"1.0","event_id":"sha256:fb7d80fd937c1bb506cfc49dda883b69b8963f800e57057130d427bb0b5eb3cf"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:CTNXVNR55V2SAHJO33PKRUFG64","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Duality and stationary distributions of the \"Immediate Exchange Model\" and its generalizations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Bart Van Ginkel, Federico Sau, Frank Redig","submitted_at":"2015-08-20T08:28:55Z","abstract_excerpt":"We prove that the \"Immediate Exchange Model\" of econophysics has a discrete dual, where the duality functions are those connecting the Brownian Energy Process and the Symmetric Inclusion Process. As a consequence, we recover invariance of products of Gamma distributions with shape parameter 2, and obtain ergodicity results. Next we show similar properties of a generalized model, where the exchange fraction is $Beta(s,t)$ distributed (instead of uniform), and product measures with $\\mbox{Gamma}(s+t)$ marginals are invariant. We prove that the discrete dual has the self-duality property, and pro"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.04918","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:18:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6umtbeGPthMWIy28ioze+NH/hWUmHMm3dzGjnd8KWX6S/ePUvQgkisj9bs3gR+KMn+6h54xyYsaIi+giKFW6Cw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T14:42:12.118851Z"},"content_sha256":"3327e8530ee00adf4d65181764463a7a0a230d42e07d13cffe774ae347d96be5","schema_version":"1.0","event_id":"sha256:3327e8530ee00adf4d65181764463a7a0a230d42e07d13cffe774ae347d96be5"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/CTNXVNR55V2SAHJO33PKRUFG64/bundle.json","state_url":"https://pith.science/pith/CTNXVNR55V2SAHJO33PKRUFG64/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/CTNXVNR55V2SAHJO33PKRUFG64/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-08T14:42:12Z","links":{"resolver":"https://pith.science/pith/CTNXVNR55V2SAHJO33PKRUFG64","bundle":"https://pith.science/pith/CTNXVNR55V2SAHJO33PKRUFG64/bundle.json","state":"https://pith.science/pith/CTNXVNR55V2SAHJO33PKRUFG64/state.json","well_known_bundle":"https://pith.science/.well-known/pith/CTNXVNR55V2SAHJO33PKRUFG64/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:CTNXVNR55V2SAHJO33PKRUFG64","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"4fb24aa8e40c893833889b078ce3e8207292f068e911c69bef6aa2e7c213c24e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-08-20T08:28:55Z","title_canon_sha256":"b59db81c6aac22f446ceadede859f0563e756d80ff94e79732086958f81615f8"},"schema_version":"1.0","source":{"id":"1508.04918","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1508.04918","created_at":"2026-05-18T01:18:43Z"},{"alias_kind":"arxiv_version","alias_value":"1508.04918v2","created_at":"2026-05-18T01:18:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1508.04918","created_at":"2026-05-18T01:18:43Z"},{"alias_kind":"pith_short_12","alias_value":"CTNXVNR55V2S","created_at":"2026-05-18T12:29:17Z"},{"alias_kind":"pith_short_16","alias_value":"CTNXVNR55V2SAHJO","created_at":"2026-05-18T12:29:17Z"},{"alias_kind":"pith_short_8","alias_value":"CTNXVNR5","created_at":"2026-05-18T12:29:17Z"}],"graph_snapshots":[{"event_id":"sha256:3327e8530ee00adf4d65181764463a7a0a230d42e07d13cffe774ae347d96be5","target":"graph","created_at":"2026-05-18T01:18:43Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We prove that the \"Immediate Exchange Model\" of econophysics has a discrete dual, where the duality functions are those connecting the Brownian Energy Process and the Symmetric Inclusion Process. As a consequence, we recover invariance of products of Gamma distributions with shape parameter 2, and obtain ergodicity results. Next we show similar properties of a generalized model, where the exchange fraction is $Beta(s,t)$ distributed (instead of uniform), and product measures with $\\mbox{Gamma}(s+t)$ marginals are invariant. We prove that the discrete dual has the self-duality property, and pro","authors_text":"Bart Van Ginkel, Federico Sau, Frank Redig","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-08-20T08:28:55Z","title":"Duality and stationary distributions of the \"Immediate Exchange Model\" and its generalizations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.04918","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:fb7d80fd937c1bb506cfc49dda883b69b8963f800e57057130d427bb0b5eb3cf","target":"record","created_at":"2026-05-18T01:18:43Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"4fb24aa8e40c893833889b078ce3e8207292f068e911c69bef6aa2e7c213c24e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-08-20T08:28:55Z","title_canon_sha256":"b59db81c6aac22f446ceadede859f0563e756d80ff94e79732086958f81615f8"},"schema_version":"1.0","source":{"id":"1508.04918","kind":"arxiv","version":2}},"canonical_sha256":"14db7ab63ded75201d2ededea8d0a6f739ea9f92b14a18851196708c8c5d6b0b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"14db7ab63ded75201d2ededea8d0a6f739ea9f92b14a18851196708c8c5d6b0b","first_computed_at":"2026-05-18T01:18:43.625535Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:18:43.625535Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"csjovNKj/maXTYACiPdxEROumBtf6bNYvqYaQjP98QoFaFKSTc9n7Kw9gLtLuoZ86bRm4Um/04tTkdexS1sPCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:18:43.626590Z","signed_message":"canonical_sha256_bytes"},"source_id":"1508.04918","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fb7d80fd937c1bb506cfc49dda883b69b8963f800e57057130d427bb0b5eb3cf","sha256:3327e8530ee00adf4d65181764463a7a0a230d42e07d13cffe774ae347d96be5"],"state_sha256":"19eb73aad738800e045772c50b5525ce892d8940a3d538fcf8f5f628982c2dd9"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6Jhpf3jbHfmw+Wu2eSKW4eGH6TNneUWNrTsUae2Kp+/MRCqZTN7tpURCbgw3EBhq8vy3LFfC5grWJ1Cul7fEAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-08T14:42:12.120769Z","bundle_sha256":"74d3398bc06f1a7673ec4f5b2f7adf45a9000dfadf71d270ccdb080252a88c81"}}